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Bell's inequality is important because it shows that quantum mechanics predicts macroscopic violations of locality. This can only be tested by suitable macroscopic measurements. To discriminate between the class of theories we are proposing one must use statistically irreversible macroscopic events to measure the timing. If the probability of reversal is sufficiently low the events can be treated as if they were absolutely irreversible. If necessary their probability of being reversed can be factored into the experimental analysis. Experimenters often implicitly assume this criteria for the completion of an event even though it cannot be justified in the formalism of quantum mechanics.
Figure: Typical experiment to test Bell's inequality
Reported experiments generally involve a setup such as that shown in Figure . Quantum mechanics predicts that the correlation between joint detection will change as a function of the polarizer (or other experimental apparatus) settings with a delay given by the time it takes light to travel the distance L. Most experiments are symmetric. L is the distance from either polarizer to the closest detector. Locality demands that a change large enough to violate Bell's inequality can only happen in the time it would take light to travel the longer distance K. K is the distance from either polarizer to the more distant detector. To show locality is violated one must show that the delay (D) between when the polarizer settings are changed and the correlations change is short enough that where C is the speed of light.
It is technically difficult to directly measure D and none of the reported experiments do this. Indirect arguments about D are all questionable. We have no idea what is happening between the time the excited state was prepared and the two detections occurred. Thus we can make no assumptions about what is happening microscopically. This is true both because quantum mechanics is silent on what is happening and because these experiments are testing the correctness of quantum mechanics itself.
To directly measure D requires that one have a high rate of singlet state events or a common trigger that controls these events and the change in polarizer angles. If this condition is not met the delay we measure will be dominated by the uncertainty in when a singlet state event occurs. After we change the parameter settings the average delay we observe will be where r is the rate of singlet state events and D is the delay we want to measure. If it will be impossible to accurately measure D. Typical experiments involve distances of a few meters. This correspond to expected values of ns. if locality holds and D < 1 ns. if quantum mechanics is correct. A high rate of singlet state events or a precise common trigger for singlet state events and changes in polarizer angles is necessary to discriminate between these times.
To show a violation of Bell's inequality one must show the superluminal transmission of information (at least by Shannon's definition of information). One must show that a change in polarizer angles changes the probability of joint detections in less time than it would take light to travel from either detector to the more distant analyzer. For this change to be sufficient to violate Bell's inequality requires that information about at least one (we cannot tell which one) polarizer setting influenced the more distant detector. There must be a macroscopic record to claim information has been transferred. It is the time of that record that must be used in determining if the information transfer was superluminal.
If one can show superluminal information transfer then one has a violation of relativistic locality (ignoring the predeterminism loophole) that is independent of the details of the experiment. Any attempt to enumerate and eliminate all loopholes is insufficient because one can never figure out all the ways that nature might out fox you.
It is worth noting that the historical roots of these predictions is the assumption that the wave function changes instantaneously when an observation occurs. This assumption has been built into the mathematics of quantum mechanics in a way that creates irreducibly nonlocal operations. Quantum mechanics insists that there is no hidden mechanistic process that enforces the conservation laws. It is this assumption that creates the singlet state entanglement that enforces conservation laws nonlocally as if by magic with no underlying mechanism.
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